Extremal elasticity of quadratic orders
Steve Fan, Paul Pollack
TL;DR
The paper analyzes the extremal elasticity $\rho(\mathcal{O}_f)$ of quadratic orders $\mathcal{O}_f$ in a fixed quadratic field, distinguishing imaginary and real cases. It develops a framework linking elasticity to the Davenport constant of (pre-)class groups via the principal part and pre-class groups, using the invariant $L(f)$ to control growth and deriving both upper and lower bounds. In the imaginary case, maximal elasticity is shown to be $\ll f$ for split-free conductors, while minimal elasticity exhibits a lower bound of $(\log f)^{c_1\log\log\log f}$ and an upper bound along a subsequence by exploiting detailed structure of $L(f)$ and prime-factor behavior. In the real case, maximal elasticity is bounded by $f/\log f$ unconditionally, with a GRH-based lower bound showing existence of infinitely many inert primes $p$ for which $\rho(\mathcal{O}_p) > p^{1/4-\varepsilon}$, and a GRH-driven result giving bounded elasticity on infinitely many split-free conductors. The methods adapt analytic-number-theory techniques used in studying multiplicative groups modulo $m$, translating them to the arithmetic of quadratic orders through the pre-class and class-group framework, and yielding precise extremal behaviors that illuminate the distribution of elasticity across conductors.
Abstract
We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log{f})^{c_1 \log\log\log{f}}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log{f})^{c_2\log\log\log{f}}$ for infinitely many $f$. Here $c_1, c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^{\times}$.